How many `3xx3` matrices M with entries from `{0, 1, 2}` are there, for which the sum of the diagonal entries of `M^(T)M` is 5 ?

Options

A. 126
B. 198
C. 162
D. 135

Answer

B

Explanation

Let `M=[a_(ij)]_(3xx3)`. Then, the diagonal elements of `M^TM` are given by
`(M^TM)_(ij)=sum_(r=1)^3(M^T)_(ir)(M)(ri)=sum_(r=1)^3a_(ri)a_(ri)=sum_(r=1) ^3(a_(ri))^2`
`=a_(1i)^2+a_(2i)^2+a_(3i)^2,i=1,2,3`
Now, `sum_(i=1)^3(M^TM)_(ij)=5`
`rArr sum_(i=1)^3(a_(1i)^2+a_(2i)^2+a_(3i)^2)=5`
`rArr` Sum of the squares of elements of M =5
We observe that that the sum of the squares of elements of M can be 5, if
(i) 5 elements of M are each equal to 1 and remaining all are zeros
Or, (ii) there is an elements equal to 1 and one element equal to 2.
So, the requiered number of matrices is `.^9C_5+^9C_1xx^8C_1=198`